We all know this, right? It's because when you skate a three turn, it looks like a figure three drawn on the ice.
However, not everyone does know this. When a coach said in a group class "We call it a three turn because it draws the pattern on the ice of a 3." a woman next to me gasped. We all instantly knew she'd just had a revelation. I said "Don't be surprised when you get to figure 8's."
So one reason things get their name is because of the shape the skate draws on the ice.
Mohawk and Choctaws
In the "History of Figure Skating" it says the names of the mohawk and choctaw to skating steps came when London skaters watching Mohawk and Choctaw dancers in a local theater observed that some of the dances or steps reminded them of skating steps.
I researched Google Books and looked at skating books from 1869 forward. In the 1869 book--no mention of mohawk or choctaw. By the 1880's mohawks and choctaws were mentioned. So those names were put on the skating moves sometime in the 1870s. (I've actually seen a book where the man who named them is named, sadly I can no longer find it).
Well, I'm going to do some speculation and see if I can make a reasonable guess of what dances or steps the skaters of the 1870's may have seen by looking at videos of Mohawk and Choctaw dances on youTube.
First, the skating choctaws of the 1870's were big, sweepy moves skated on pond ice. They may have resembled a choctaw as demonstrated by Maya Usova. It's a large serpentine move.
I couldn't find a step like that, but I could find a traditional Choctaw dance that had a serpentine shape so perhaps this dance form inspired the name of the choctaw step.
Everyone learns a mohawk early on in their skating life so we all know what it looks like. There's a shift from one foot to the other, and a shift from forward moving to back motion.
In this demo we see a skating mohawk. I coudn't find an exact match to that. But there are lots of mohawk 'tricks' including mohawk barrel rolls (thank you Coach Bianca for your mohawk obsession).
So let's look at a mohawk barrel roll (sorry about the hockey skates).
And now we're getting somewhere. I can't match the big sweepy mohawk, but a step like a mohawk barrel roll does appear in several Mohawk dances.
Below follow the dancer in the large white head dress to see the use of the step that reminds me of a skating mohawk barrel roll.
So there's your figure skating history speculation for the day!